The basic symmetry structural unit for description of non-crystallographic symmetry of hydrocarbon-like chains

^{1}Rabinovich A.L., ^{2}Talis A.L.

^{1}IKarelian Research Centre of Russian Academy of Sciences ^{2}Institute of organoelement compounds RAS

By analogy with the tetrahedral chains, the basic symmetry structural unit for tetracoordinated chains is revealed. It is a part of mapping of the polytope {240} substructures into the 3-dimensional Euclidean space, and is a combined tetrablock which realized as (i) a 14-vertex “composite” tetrablock version and (ii) an 11-vertex “decorated” tetrablock version.

Description of non-crystallographic symmetry of hydrocarbon chains of biomembrane phospholipids

^{1}Rabinovich A.L., ^{2}Talis A.L.

^{1}IKarelian Research Centre of Russian Academy of Sciences ^{2}Institute of organoelement compounds RAS

It is shown that a 14-vertex and an 11-vertex tetracoordinated structures which are part of the 4-dimensional ideal polyhedron (the polytope {240}) mapping into the 3-dimensional Euclidean space, allow the non-crystallographic symmetry of linear hydrocarbon chains - typical components of biomembrane phospholipids to be displayed.

Tetrablock helices with non-crystallographic symmetry

^{1}Talis A.L., ^{2}Rabinovich A.L.

^{1}Institute of organoelement compounds RAS ^{2}Karelian Research Centre of Russian Academy of Sciences

The symmetry structural unit of a tetrahedral chain is a seven-vertex linear union of four face-sharing regular tetrahedral (a tetrablock). When combining two tetrablocks over end faces, there may be three versions differing in the turning angle (clockwise in the plane of common face) between the second tetrablock and the first one (0^{o}, 120^{o}, 240^{o}). So, connecting uniformly linear versions of tetrablocks of the same chirality over end faces allows obtaining three versions of “tetrablock helices”; these helices are described in this work.

The Hopf fibration and two variants of the polytope {240} decomposition into linear substructures

^{1}Talis A.L., ^{2}Rabinovich A.L.

^{1}Institute of organoelement compounds RAS ^{2}Karelian Research Centre of Russian Academy of Sciences

A polyhedron in 4-dimensional Euclidean space of constant positive curvature is considered (polytope {240}) consisted of 240 vertices which are tetra coordinated. There is a known option for mapping a linear substructure of the polytope {240} into a channel in 3-dimensional Euclidean space. The choice of a subsystem of the polytope {240} edges that includes all its vertices is not unique; other options are possible for choosing linear substructures of the polytope. In this work, we propose a variant in which sets of vertices form chain sequences, and not channels